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State-identification Experiments and Testing of Sequential Circuits 1 Zvi Kohavi and Niraj K. Jha Experiments Experiment: application of an input sequence to the input terminals of a machine • Simple: performed on a single copy • Multiple: performed on two or more identical copies • Adaptive: input symbol at any time instant depends on previous output symbols • Preset: entire input sequence is predetermined • Length: total number of symbols in the experiment Checking experiment: designed to take the machine through all its transitions to ascertain if it is working correctly 2 Introductory Example Example: Consider machine M1 and its responses to 01 and 111 • Output response to 01: uniquely determines the machine’s final state, but not the initial state • Output response to 111: uniquely determines the machine’s final and initial states 3 Uncertainties Initial uncertainty: minimal subset of set of states S, which is known to contain the initial state • For machine M1: (ABCD) If application of input symbol 1 results in output symbol 1: uncertainty is (ACD) • • • • 1-successors of (ACD): uncertainty vector (B)(CD) Components of uncertainty vector: (B) and (CD) Trivial uncertainty vector: all components have a single state Homogeneous uncertainty vector: components contain a single state or identical repeated states 4 Successor Tree Successor tree: displays Ii-successor uncertainties for all Ii • Path: sequence of j branches starting at the highest node and terminating at the jth level • Length of the path: j • Each path describes: an input sequence Level (ABCD) 0 0 1 (A)(BCC) (ACD)(B) 0 0 (AA)(C)(C) 1 (A)(BB)(D) (A)(BC)(C) 0 1 (A)(A)(C)(C) (A)(B)(B)(D) 1 1 (A)(B)(CD) 0 2 1 (A)(B)(C)(C) (A)(B)(C)(D) 3 5 Homing Experiments An input sequence Y0 is a homing sequence if the final state of the machine can be determined uniquely from the machine’s response to Y0, regardless of the initial state Homing tree: successor tree in which a jth-level node becomes terminal when: 1. It is associated with an uncertainty vector whose non-homogeneous components are associated with some node in a preceding level 2. It is associated with a trivial or homogeneous vector Example: Consider machine M2 Level (ABCD) 0 1 (AB)(DD) 0 (AB)(DD) 0 (ABCD) 1 1 (BD)(CC) 0 (A)(D)(DD) 2 1 (AA)(BC) 3 Theorem: A preset homing sequence of length at most (n-1)2 exists for every reduced n-state machine 6 Synchronizing Experiments A synchronizing sequence of a machine M is an input sequence that takes M to a specified final state, regardless of the outputs symbols or the initial state In a synchronizing tree: a jth-level node becomes terminal whenever: 1. The node is associated with an uncertainty that is also associated with some node in a preceding level 2. Some node in the jth level is associated with an uncertainty containing just a single element Level Example: Machine M2 and its synchronizing tree (ABCD) 0 1 (ABD) 0 (ABCD) 1 1 (ABD) (BCD) 0 0 2 1 (AD) Theorem: If a synchronizing sequence for an n-state machine M exists, then its length is at most (n-1)2n/2 0 (ABC) 3 1 (BD) (CD) 0 4 1 7 (D) (AC) 5 Distinguishing Experiments An input sequence X0 of machine M is said to be a distinguishing sequence if the resulting output sequence is different for each initial state • Since the knowledge of the initial state and input sequence is always sufficient to uniquely determine the final state: every distinguishing sequence is also a homing sequence – The converse is not true Distinguishing tree: successor tree in which a node at the jth level becomes terminal when: 1. The node is associated with an uncertainty vector whose nonhomogeneous components are associated with some node in a preceding level 2. The node is associated with an uncertainty vector containing a homogeneous nontrivial component 3. Some node at the jth level is associated with a trivial uncertainty vector 8 Distinguishing Experiments (Contd.) Example: Machine M1 and its distinguishing tree Level (ABCD) 0 0 1 (A)(BCC) (ACD)(B) 0 0 (AA)(C)(C) 1 (A)(BB)(D) 1 (A)(BC)(C) 0 1 (A)(B)(CD) 1 0 (A)(A)(C)(C) (A)(B)(B)(D) 2 1 (A)(B)(C)(C) (A)(B)(C)(D) 3 Example: While every machine has a homing sequence, not every machine has a distinguishing sequence – Consider M2 and its distinguishing tree Level (ABCD) 0 1 (AB)(DD) 0 (AB)(DD) 0 (ABCD) 1 1 (BD)(CC) 0 2 1 9 (A)(D)(DD) (AA)(BC) 3 Shortest Distinguishing Prefix Example: Consider machine M1 and its response to 111 • Shortest distinguishing prefix for state C: 1 • For state D: 11 • For states A and B: 111 10 Machine Identification Machine identification: experimentally determining the state table of an unknown machine • Input alphabet known • Upper bound on the number of states known • Machine must be reduced and strongly connected Example: Suppose a machine is supposed to have two states and its response to input sequence X is output sequence Z: • Corresponding machine M3: 11 Checking Experiments Given a machine and its state table: determine from terminal experiments whether the actual machine is isomorphic to the one described by the state table • • • • Machine assumed to be: strongly connected, completely specified and reduced Faults assumed to be: permanent Machine either has a synchronizing sequence: or a reset input exists that will transfer it to the initial state Initial assumption: a distinguishing sequence exists Designing checking experiments: two parts 1. Use the synchronizing sequence or reset input to transfer the machine into a prespecified state: initial state for the second part of the experiment 2. Preset experiment: take machine through all possible transitions 1. Machine is caused to display the response of each of its states to the distinguishing sequence 2. Then, actual state transitions are verified 12 Example Example: Machine M4 and its responses to distinguishing sequences 00 and 01 • Suppose the synchronizing sequence or reset input places M4 into A • First ascertain: starting state is indeed A and the machine being tested actually contains four distinct states • Next: verify every state transition – Apply the input symbol corresponding to the transition – Identify it by applying the distinguishing sequence 13 Example (Contd.) Example (contd.): First stage for checking all state transitions except those from D to A and A to B and C 14 Example (Contd.) Example (contd.): Complete experiment: 15 Design of Diagnosable Machines Diagnosable machine: one which possesses one or more distinguishing sequences Testing table and graph: for machine M2 that does not possess a distinguishing sequence 0/0 AB 1/0 BC AC 1/0 BD 1/0 1/0 AD 1/0 1/0 CD Uncertainty pair Implied pair 16 Definitely Diagnosable Machines A machine is definitely diagnosable machine of order if is the least integer s.t. every sequence of length is a distinguishing sequence for M • Every node in level of the distinguishing tree is associated with a trivial uncertainty vector Theorem: A machine is definitely diagnosable if and only if its testing graph G is loop-free and no repeated states exist in the testing table Corollary: Let the testing table of machine M be free of repeated entries, and let G be a loop-free testing graph for M • If the length of the longest path in G is l: then = l + 1 17 Designing Definitely Diagnosable Machines Example: Obtaining definitely diagnosable M2’ from M2 • Assign different output symbols to each transition that may cause a repeated entry in the testing table • Open loops in the testing graph by removing one of the arcs 0/0 AB 1/0 BC AC 1/0 BD 1/0 1/0 AD 1/0 1/0 CD 18 Definitely Diagnosable Machines (Contd.) For any 2k-state machine: addition of k output terminals is sufficient to convert it into a definitely diagnosable machine x Logic z Logic z Logic z1 S M (a) Machine M. x S M (b) Machine M . z1 is only used for diagnosis purposes. 19 State Table based Test Generation Functional fault model: faults assumed to be associated with a state transition • • • • • Single-state-transition (SST) fault model: fault results in the destination state of the state transition becoming corrupted while retaining its correct input/output symbols Test generation based on SST faults: known to detect a very high percentage of single stuck-at faults in the sequential circuit Assumption: SST fault does not increase the number of states in the state table State transition designated as a four-tuple: <input symbol, source state, destination state, output symbol> A state transition can become corrupted: if its destination state, output symbol or both are faulty – However, if a test sequence detects a corrupted destination state: then it also detects the corrupted output symbol or both the corrupted destination state and output symbol – Three parts to a test sequence: » Initialization sequence » Input symbol of the transition to activate the fault » State-pair differentiating sequence (SPDS) 20 Test Generation (Contd.) Fault collapsing: An n-state m-transition machine has m(n-1) SST faults • • • For each state transition: there are n-1 faulty destination states possible Suppose the four-tuple <Ik,Sj,Si,Ol> is corrupted to: <Ik,Sj,Si’,Ol> by SST fault f1 and to <Ik,Sj,Si’’,Ol> by SST fault f2 If SPDS(Si,Si’) also differentiates between Si and Si’’: fault f2 dominates fault f1, and f2 can be removed from the fault list Example: Consider machine M5 • • • SPDS(A,B) = SPDS(A,C) = 0 and SPDS(B,C) = 1 Consider <1,C,A,0>: can be corrupted to <1,C,B,0>, <1,C,C,0>, <1,C,D,0> Since SPDS(A,B) = SPDS(A,C): the first two faulty transitions can be collapsed into just the first one 21 Test Generation Example Example: Consider the SST fault that corrupts <0,D,A,0> to <0,D,B,0> • • • • We first need transfer sequence T(A,D) = 00 (10 is also a valid sequence) Fault is activated by x = 0 Finally, SPDS(A,B) = 0 is applied Hence, one possible test sequence: 0000 22 Sequential Circuit based Test Generation Extended D-algorithm: 1. Target a fault in some time frame, say time frame 0 in the iterative array model: use D-algorithm to generate a test vector for it 2. If a D or D’ propagates to a circuit output: no further error propagation required 3. If D or D’ only propagates to next state lines: add new time frames to the right until the error signal reaches some circuit output 4. If test vector contains assignments of specific logic values to any present state lines in time frame 0: add new time frames to the left until no particular values are required on the present state lines 23 Example Example: Test sequence for x2 s-a-1: {(1,1),(1,0),(1, )} z x1 x2 s-a-1 D y Y (a) A sequential circuit. z0 z -1 x1-1 1 x2-1 1 s-a-1 -1 y x10 1 D x20 s-a-1 0 Y Time frame -1 -1 D 1 y0 Time frame 0 D 0 Y z1 D x11 1 x21 s-a-1 Y y1 1 Time frame 1 (b) Iterative array model. 24 Nine-valued Logic Five-valued logic {0,1,,D,D’} used in D-algorithm: not adequate for sequential circuits because it overspecifies the value requirements at some lines in the circuit • This may prevent the test generator from obtaining a test sequence even when one exists • This problem can be tackled by nine-valued logic: 0/0, 0/1, 0/, 1/0, 1/1, 1/, /0, /1, / 25 Example Example: Unsuccessful test generation with five-valued logic x1 s-a-1 z x2 y D Y (a) A sequential circuit. x1-1 s-a-1 0 x2-1 0 y -1 conflict x10 G1 s-a-1 D 0 z -1 1 Y Time frame -1 -1 0 D 0 x20 1 1 D z0 Y 0 0 y 0 Time frame 0 (b) Application of the extended D-algorithm. 26 Example (Contd.) Example (contd.): Successful test generation with nine-valued logic • x1-1 s-a-1 x10 0/ x2-1 0/ y -1 Test sequence: {(0,0),(0,1)} 0/1 z -1 1/ Y Time frame -1 -1 0/ G1 s-a-1 x20 1/ 0/ 0/1 0/0 0/ y0 1/0 1/ z0 Y 0 Time frame 0 27 Design for Testability Scan design for sequential circuits: two modes of operation • • • • Normal mode: circuit exhibits original behavior Test mode: its flip-flops are chained together into a shift register Full-scan: all flip-flops are chained Partial-scan: a subset of flip-flops are chained Scan flip-flop and its symbol: Yi M U X S Yi M U X Yi yi D S Yi yi D T T (a) Scan flip-flop. (b) Symbol. Scan chain: only combinational test generation needed x1 z1 x2 xl z2 Combinational logic y1 Y2 Y1 ScanIn YS 1 T M U X y2 D YS 2 M U X yk Yk D S Y3 YS k M U X zm D ScanOut 28 Scan-based Test Application Application of the test set derived for the combinational logic to the sequential circuit: 1. Make T = 1 to set circuit into the test mode 2. Scan in the state part of the vector through the ScanIn input in the next k clock cycles. Primary inputs can be fed arbitrary values in these cycles 3. Apply the primary input part of the vector to the primary inputs. Now, all l+k bits of the test vector have been applied to the combinational logic. After allowing the logic to settle down, observe the output response at circuit outputs z1, z2, …, zm 4. Make T = 0 to set the circuit into the normal mode 5. Apply a clock pulse. This results in the values on the next state lines, Y1, Y2, …, Yk, being latched in the k flip-flops 6. Make T = 1 and observe the values captured in the flip-flops by scanning them out through ScanOut while repeating the procedure for the next test vector 29 Testing of Scan Designs Suppose there are n test vectors in the test set • • • • A total of k cycles required to scan in the state part One cycle to capture state response k-1 cycles required to scan out the captured state Since the state part of the next state vector is scanned in at the same time the captured state for the previous vector is being scanned out: total no. of clock cycles required for testing = n(k+1)+k-1 Example: total no. of clock cycles = 4(2+1)+2-1 = 13 y2 y1 z x1 z x1 y2 Y1 x1 1 0 1 0 x2 0 1 1 0 y1 0 1 0 1 y2 1 0 0 1 x2 y1 x2 Y2 y1 D y2 D (a) Sequential circuit. Y1 Y2 (b) Combinational logic. (c) Stuck-at fault test set. 30 Built-in Self-test (BIST) Integration of circuitry on-chip to enable the circuit under test (CUT) to test itself • • • • At-speed testing possible: at the normal clock rate – Detects delay faults Test pattern generator (TPG) Response analyzer (RA): compresses output response into a signature Golden signature: when no fault is present T P G CUT R A 31 Test Pattern Generator TPG: usually a linear feedback shift register (LFSR) • LFSR with degree-k feedback polynomial: p(x) = xk + b1xk-1 + … + bk-1x + bk Yk • b1 bk-1 bk yk + D Y2 Yk-1 y2 + D Y1 y1 D Outputs of the k flip-flops directly fed to inputs of a k-input CUT Example: Three-stage LFSR with p(x) = x3 + x2 + 1 Y3 D y3 Y2 y2 D + Y1 y1 D 32 Feedback Polynomial A feedback polynomial is said to be primitive if the state diagram corresponding to the k-stage LFSR consists of two loops: • • Trivial loop: with the all-0 state Non-trivial loop: with the remaining 2k-1 states Example: State diagram of the three-stage LFSR based on p(x) = x3+x2+1 • Thus, p(x) is primitive 111 101 110 001 011 010 100 000 33 LFSR Seed LFSRs based on primitive polynomials find use in BIST • • • Test pattern can start with any state in the non-trivial loop Initial state: seed LFSR re-seeding: Start from different seeds and apply a few test patterns from each in order to shorten the test application time Example: For the circuit below, possible stuck-at test set: (x3,x2,x1) = {(1,0,1), (1,1,1), (1,0,0), 0,1,0)} • • Choose three-stage LFSR: yi connected to xi Testing accomplished by: two patterns starting with seed (1,0,1) and two additional patterns starting with (1,0,0) 111 – Alternative: six clock cycles from (1,0,1) to (0,1,0) 110 101 x1 x3 x2 z Y3 D y3 Y2 y2 D + Y1 y1 D 001 011 010 100 000 34 Response Analyzer For a k-output CUT, to which n patterns have been applied by the TPG: need to analyze kn output bits to detect error • • Storing these bits and performing bit-by-bit comparison to error-free values is expensive in space and time: – Thus, RAs used to compress the output response Aliasing: Signature in the presence of a fault is the same as the golden signature – Typically, negligible aliasing probability = 1/2m zk-1 zk + bk D + bk-1 z1 D + D b1 Multiple-input signature register 35